Today's word isn't so much a word as a sound: dai, pronounced – approximately – 'dye-yuh!'
Luckily it's simpler to translate than it is to transcribe phonetically: it means something like 'come on'.
Come on, let's go!
It's beautifully versatile for one little word. You can yell it at your favourite team to encourage them (or to tell them they need to do better), whisper it to a friend to invite them to go ahead, or whine it pleadingly to get someone to give in.
Come on, hurry up!
Dai, resta ancora un po'.
Go on, stay a bit longer.
Dai, dai, ti prego Papà!
Oh go on, go on, please Dad!
Its various tones are hard to pinpoint, but according to one Italian dictionary, adding 'but' in front of it makes it sound incredulous…
– Io preferisco la pizza americana.
– Ma dai!
– I prefer American pizza.
– Oh come on!
… while prefacing it with 'and' indicates irritation and impatience.
– Non è giusto però!
– E dai! Smettila.
– But it's not fair!
– Come on! Stop that.
However you use dai, do it with conviction. You can stretch out its vowels as long as you dare, and if you wish to repeat it a few times for emphasis or accompany it with an exasperated flourish of the hand, well, we won't judge you. Daaaaaaiii!